Since the phrase is so ambiguous, it's hard to give a direct example. But if we assume it means believing in big or wild stories, an example could be an old woman in a village believing in a story about a giant rooster that could talk and fly like a bird of prey. But this is just an example based on a very uncertain interpretation.
Let's say there's a community where people often tell exaggerated stories. The toothless old woman is part of this community. One day, someone tells her a story about a magical rooster ('cock') that has feathers made of gold and can lay eggs that turn into precious jewels. Despite how unbelievable it sounds, she 'swallows' this story, meaning she believes it. This is a very loose example based on trying to make sense of this rather strange phrase.
Given the vagueness of the statement, we could perhaps consider a scenario where an old woman with no teeth is told a story about a rooster as big as a house and she just accepts it without question. So, for instance, in a fictional folktale - like setting, the toothless old woman is told that this huge 'cock' (rooster) saved the village once by scaring away all the wolves with its giant size and she swallows this story whole, believing it to be true.